4.2 Convection

As we discussed at the beginning of Section 3, it is becoming increasingly accepted that convection
above the proto neutron star plays a major role in the supernova explosion mechanism (e.g.,
Figure 1). Convectively-driven inhomogeneities in the density distribution of the outer regions of
the nascent neutron star and anisotropic neutrino emission are other sources of GW emission
during the collapse/explosion [37, 206]. Inside the proto neutron star, GW emission from these
processes results from small-scale asphericities, unlike the large-scale motions responsible for GW
emission from aspherical collapse and non-axisymmetric global instabilities. But the SASI and
low-mode convection do produce large-scale accelerated mass motions [165, 213]. Note that
Rayleigh–Taylor instabilities in the exploding star also induce time-dependent quadrupole moments at
composition interfaces in the stellar envelope. However, the resultant GW emission is too weak
to be detected because the Rayleigh–Taylor instabilities occur at very large radii (with low
velocities) [206].

4.2.1 Asymmetric collapse

Since convection was suggested as a key ingredient in the explosion, it has been postulated that
asymmetries in the convection can produce the large proper motions observed in the pulsar
population [139]. Convection asymmetries can either be produced by asymmetries in the progenitor star
that grow during collapse or by instabilities in the convection itself. Burrows and Hayes [37] proposed that
asymmetries in the collapse could produce pulsar velocities. The idea behind this work was that
asymmetries present in the star prior to collapse (in part due to convection during silicon and oxygen
burning) will be amplified during the collapse [13, 172]. These asymmetries will then drive
asymmetries in the convection and ultimately, the supernova explosion. Burrows and Hayes [37] found
that not only could they produce strong motions in the nascent neutron star, but detectable
GW signals. The peak amplitude calculated was , for a source located at
10 Mpc.

Fryer [98] was unable to produce the large neutron star velocities seen by Burrows and Hayes [37] even
after significantly increasing the level of asymmetry in the initial star in excess of 25%. More recent results
by Burrows’ group [212] suggest that the Burrows and Hayes results were not quantitatively correct.
However, the GW signal produced by the more recent and the original simulations is comparable. Figure 17
shows the gravitational waveform from the Burrows & Hayes simulation (including separate matter and
neutrino contributions). Note that the neutrinos dominate the amplitude of the GW signal. As we shall see
in Section 4.4, the neutrino contribution may dominate the GW signal from many asymmetric collapse
simulations. Figure 18 shows the matter contribution to the gravitational waveforms for the Fryer
results.

Figure 17: The gravitational waveform (including separate matter and neutrino contributions) from
the collapse simulations of Burrows and Hayes [37]. The curves plot the GW amplitude of the source
as a function of time. (Figure 3 of [37]; used with permission.)

Figure 18: The gravitational waveform for matter contributions from the asymmetric collapse
simulations of Fryer et al. [107]. The curves plot the GW amplitude of the source as a function of
time. (Figure 3 of [107]; used with permission.)

The study by Nazin and Postnov [220] predicts a lower limit for emitted during an asymmetric
core-collapse SN (where such asymmetries could be induced by both aspherical mass motion and neutrino
emission). They assume that observed pulsar kicks are solely due to asymmetric collapse. They suggest that
the energy associated with the kick (, where and are the mass and velocity of the neutron
star) can be set as a lower limit for (which can be computed without having to know the
mechanism behind the asymmetric collapse). From observed pulsar proper motions, they estimate
the degree of asymmetry present in the collapse and the corresponding characteristic GW
amplitude (). This amplitude is 3 × 10–25 for a source located at 10 Mpc and emitting at
.

4.2.2 Proto neutron star convection

Müller and Janka [208] performed both 2D and 3D simulations of convective instabilities in the proto
neutron star and hot bubble regions during the first second of the explosion phase of a Type II SN.
They numerically computed the GW emission from the convection-induced aspherical mass
motion and neutrino emission in the quadrupole approximation (for details, see Section 3 of their
paper [208]).

Figure 19: Convective instabilities inside the proto neutron star in the 2D simulation of Müller and
Janka [208]. The evolutions of the temperature (left panels) and logarithmic density (right panels)
distributions are shown for the radial region 15 – 95 km. The upper and lower panels correspond to
times 12 and 21 ms, respectively, after the start of the simulation. The temperature values range
from 2.5 × 1010 to 1.8 × 1011 K. The values of the logarithm of the density range from 10.5 to
13.3 g cm–3. The temperature and density both increase as the colors change from blue to green,
yellow, and red. (Figure 7 of [208]; used with permission.)

For typical iron core masses, the convectively unstable region in the proto neutron star extends over the
inner of the core mass (this corresponds to a radial range of 10 – 50 km). The
convection in this region, which begins approximately 10 – 20 ms after the shock forms and may last for
20 ms – 1 s, is caused by unstable gradients in entropy and/or lepton number resulting from the
stalling of the prompt shock and deleptonization outside the neutrino sphere. Müller and Janka’s
simulations of convection in this region began with the 1D, non-rotating, 12 ms post-bounce model of
Hillebrandt [142]. This model included general-relativistic corrections that had to be relaxed away prior to
the start of the Newtonian simulations. Neutrino transport was neglected in these runs (see Section 2.1
of [208] for justification); however, a sophisticated equation of state was utilized. Figure 19 shows the
evolution of the temperature and density distributions in the 2D simulation of Müller and
Janka.

The peak GW amplitude resulting from convective mass motions in these simulations of the proto
neutron star was 3 × 10–24 in 2D and 2 × 10–25 in 3D, for d = 10 Mpc. More recent
calculations get amplitudes of 10–26 in 2D [209] and 3 – 5 × 10–26 in 3D [107]. The emitted
energy was 9.8 × 1044 erg in 2D and 1.3 × 1042 erg in 3D. The power spectrum peaked at frequencies of
200 – 600 Hz in 2D and 100 – 200 Hz in 3D. Such signals would not be detectable with LIGO-II. The
reasons for the differences between the 2D and 3D results include smaller convective elements and less
under and overshooting in 3D. The relatively low angular resolution of the 3D simulations
may also have played a role. The quadrupole GW amplitude from the 2D simulation is
shown in the upper left panel of Figure 20 (see [350, 306] for expressions relating to
).

Figure 20: Quadrupole amplitudes [cm] from convective instabilities in various models
of [208]. The upper left panel is the amplitude from a 2D simulation of proto neutron star convection.
The other three panels are amplitudes from 2D simulations of hot bubble convection. The imposed
neutrino flux in the hot bubble simulations increases from the top right model through the bottom
right model. (Figure 18 of [208]; used with permission.)

4.2.3 Convection above the proto neutron star

Convection in the hot bubble region between the shock and neutrino sphere arises because of an unstable
entropy gradient resulting from the outward moving shock and subsequent neutrino heating. Figure 21
shows a movie of the development of this entropy-driven convection. This unstable region extends over the
inner mass range (corresponding to a radial range of 100 – 1000 km). Convection in
the hot bubble begins 50 – 80 ms after shock formation and lasts for 100 – 500 ms. Only 2D
simulations were performed in this case. These runs started with a 25 ms post-bounce model provided by
Bruenn [32]. A simple neutrino transport scheme was used in the runs and an imposed neutrino flux was
located inside the neutrino sphere. Due to computational constraints, the computational domain did not
include the entire convectively unstable region inside the proto neutron star (thus, this set of
simulations only accurately models the convection in the hot bubble region, not in the proto neutron
star).

Figure 21: gif-Movie (15979 KB)
Isosurface of material with radial velocities of 1000 km s–1 for three different simulation
resolutions. The isosurface outlines the outward moving convective bubbles. The open spaces mark
the downflows. Note that the upwelling bubbles are large and have very similar size scales to the
two-dimensional simulations. From Fryer & Warren [112].

The peak GW amplitude resulting from these 2D simulations of convective mass motions in the hot
bubble region was , for . The emitted energy was . The
energy spectrum peaked at frequencies of 50 – 200 Hz. As the explosion energy was increased (by increasing
the imposed neutrino flux), the violent convective motions turn into simple rapid expansion. The resultant
frequencies drop to . The amplitude of such a signal would be too low to be detectable with
LIGO-II.

The case for GWs from convection-induced asymmetric neutrino emission has also varied with time.
Müller and Janka estimated the GW emission from the convection-induced anisotropic neutrino
radiation in their simulations (see [208] for details). They found that the amplitude of the GWs
emitted can be a factor of 5 – 10 times higher than the GW amplitudes resulting from convective
mass motion. More recent simulations by Müller et al. [209] argue that the GWs produced by
asymmetric neutrino emission is weaker than that of the convective motions. But Marek et
al. [196] have argued strongly that calculating the GW signal from asymmetries in neutrinos is
extremely difficult and detailed neutrino transport is required to determine the GW signal from
neutrinos.

Our understanding of the convective engine is deepening with time. The latest focus of attention has been
the SASI instability, which produces, at late times, extremely low mode convection. This topic is currently a
matter of heated debate. Whether this instability dominates at late times, or whether the late-time,
low-mode convection is simply the merger of convective cells [139] is, in the opinion of these authors, yet to
be conclusively determined. Convection is very difficult to simulate and has been studied for many decades
on a variety of applications from the combustion engine to astrophysics, with no accepted resolution. But
many groups are now finding low-mode convection above the proto neutron star (which can heighten the
GW emission) and all agree that this occurs at late times (more than a few hundred milliseconds
after bounce). As we have shown in Figure 5, such late explosions will be weak and, if the
assumptions of that analysis are correct, these late explosion mechanisms cannot explain the observed
core-collapse supernovae. Marek & Janka [195] believe otherwise. Certainly, if material can
continue to accrete onto the neutron star after the launch of the explosion, which it does in
some of the recent results of the Mezzacappa team (in preparation), stronger explosions may be
produced.

In addition, Burrows et al. [40] found that the downstreams in the SASI can drive oscillations in the
neutron star, which may also be a source for GWs (see Figure 22). Many groups have shown that
the convection does not, and can not in semi-analytic studies, excite the strong oscillations
observed [116, 343, 195], but see Weinberg & Quataert [327]. When low-mode convection does occur as in
the SASI, a GW signal is produced and has been modeled by several groups [85, 229, 228, 196, 165]. Most
find that the actual matter signal is quite weak: , for , but the
SASI can drive oscillations in the neutron star that produce a potentially important neutrino
signal.